More recently, a variational approach has been proposed by Lin and Wang for damping motion with a Lagrangian holding the energy term dissipated by a friction force. However, the modified Euler-Lagrange equation obtained within their formalism leads to an incorrect Newtonian equation of motion due to the nonlocality of the Lagrangian. In this communication, we generalize this approach based on the fractional actionlike variational approach and we show that under some simple restrictions connected to the fractional parameters introduced in the fractional formalism, this problem may be solved.

For a polynomial $p(z)$ of degree $n$ which has no zeros in $|z|< 1$, Dewan et al., (K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 (2010), 38-41) established$$\Big|zp'(z)+\frac{n\beta}{2}p(z)\Big|\leq\frac{n}{2}\Big\{\Big(\Big|\frac{\beta}{2}\Big|+\Big|1+\frac{\beta}{2}\Big|\Big)\max_{|z|=1}|p(z)|-\Big(\Big|1+\frac{\beta}{2}\Big|-\Big|\frac{\beta}{2}\Big|\Big)\min_{|z|=1}|p(z)|\Big\},$$for any complex number $\beta$ with $|\beta|\leq 1$ and $|z|=1$. In this paper we consider the operator $B$, which carries a polynomial $p(z)$ into$$B[p(z)]:=\lambda_0p(z)+\lambda_1\Big(\frac{nz}{2}\Big)\frac{p'(z)}{1!}+\lambda_2\Big(\frac{nz}{2}\Big)^2\frac{p''(z)}{2!}, $$ where $\lambda_0,$ $\lambda_1$, and $\lambda_2$ are such that all the zeros of $u(z)=\lambda_0+c(n,1)\lambda_1z+c(n,2)\lambda_2z^2$ lie in the half plane $|z|\leq |z-{n}/{2}|.$ By using the operator $B$, we present a generalization of result of Dewan. Our result generalizes certain well-known polynomial inequalities.

In this paper we consider the class of Bazilevic functions for bi-univalent functions. For this we will estimate the coefficients $a_2$ and $a_3$ using Caratheodory functions and the method of differential subordination.

In this paper we propose the $q$ analogues of modified Baskakov-Szász operators. We estimate the moments and establish direct results in terms of modulus of continuity. An estimate for the rate of convergence and weighted approximation properties of the $q$ operators are also obtained.

In this paper we consider a class of polynomials $P(z)= a_{0} + \sum_{v=t}^{n} a_{v}z^{v}$, $t\geq 1,$ not vanishing in $|z|<k,$ $ k\geq 1$ and investigate the dependence of ${\max_{|z|=1}}|P(Rz)-P(rz)|$ on ${\max_{|z|=1}}|P(z)|,$ where $ 1 \leq r < R.$ Our result generalizes and refines some known polynomial inequalities.

In this paper we establish sharp Hölder estimates of harmonic functions on a class of connected post critically finite (p.c.f.) self-similar sets, and show that functions in the domain of Laplacian enjoy the same property. Some well-known examples, such as the Sierpinski gasket, the unit interval, the level $3$ Sierpinski gasket, the hexagasket, the $3$-dimensional Sierpinski gasket, and the Vicsek set are also considered.

Let $P(z)$ be a polynomial of degree $n$ and for any complex number $\alpha$, let $D_{\alpha}P(z)=nP(z)+(\alpha -z)P'(z)$ denote the polar derivative of $P(z)$ with respect to $\alpha$. In this paper, we obtain certain inequalities for the polar derivative of a polynomial with restricted zeros. Our results generalize and sharpen some well-known polynomial inequalities.

In this paper, We study a general class of nonlinear degenerated elliptic problems associated with the differential inclusion $\beta(u)-div(a(x,Du)+F(u))\ni f$ in $\Omega $, where $f\in L^{1}(\Omega )$. A vector field $a(\cdot,\cdot)$ is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for general $L^1$-data. Under an additional strict monotonicity assumption uniqueness of the renormalized solution is established.